DOCSLIB.ORG

Strongly Exponential Symmetric Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Strongly Exponential Symmetric Spaces STRONGLY EXPONENTIAL SYMMETRIC SPACES YANNICK VOGLAIRE Abstract. We study the exponential map of connected symmetric spaces and characterize, in terms of midpoints and of infinitesimal conditions, when it is a diffeomorphism, generalizing the Dixmier–Saito theorem for solvable Lie groups. We then give a geometric characterization of the (strongly) exponential solvable symmetric spaces as those spaces for which every triangle admits a unique double triangle. This work is motivated by Weinstein’s quantization by groupoids program applied to symmetric spaces. 1. Introduction In the late 1950’s, Dixmier [5] and Saito [16] characterized the strongly exponential Lie groups: the simply connected solvable Lie groups for which the exponential map is a global diffeomorphism. They are characterized by the fact that the exponential map is either injective or surjective, or equivalently the fact that the adjoint action of their Lie algebra has no purely imaginary eigenvalue. In contrast, the exponential map of semisimple Lie groups is never injective, but may be surjective or have dense image. As a result, the latter two weaker notions of exponentiality have been studied for general Lie groups by a number of authors (see [6, 21] for two surveys and for references). Recently, a number of their results were generalized to symmetric spaces in [15]. The first main result of the present paper intends to complete the picture by giving a version of the Dixmier–Saito theorem for symmetric spaces (see Theorems 1.1, 1.2 and 1.3). The second main result gives a geometric characterization of the strongly ex- arXiv:1303.5925v2 [math.DG] 6 Apr 2014 ponential symmetric spaces which are homogeneous spaces of solvable Lie groups, in terms of existence and uniqueness of double triangles (see Theorem 1.4). This Part of this work was done during the author’s PhD thesis at the Université Catholique de Louvain, Belgium, and the Université de Reims, France, under the supervision of Pierre Bieliavsky and Michael Pevzner. The author thanks Mathieu Carette for discussions on midpoints and geodesics in symmetric spaces. This work was partially supported for travel by the Belgian Science Policy (belspo) through the Interuniversity Attraction Pole (IAP) PHASE VII/18 project “DYGEST”. The published version of this paper is available at http://imrn.oxfordjournals.org/cgi/ content/abstract/rnt149?ijkey=My6gILrUCEG3Z1z&keytype=ref. 1 2 YANNICK VOGLAIRE is related to a beautiful theorem of A. Weinstein [20] relating the graph of multi- plication in the pair groupoid of a symplectic symmetric space to the symplectic area of such double triangles. Weinstein’s theorem was developed in the context of his quantization by groupoids program and forms, together with the extension of Bieliavsky’s quantization of symplectic symmetric spaces [3], the initial motivation for the present work. Since we will only be concerned with strong exponentiality in this paper, from now on we will drop the adjective “strong”, although nowadays exponentiality has a weaker meaning in the literature. A symmetric space is a manifold M endowed with a family of involutions {sx}x∈M of M, called symmetries, such that x is an isolated (not necessarily unique) fixed point of sx, and such that sx(syz) = s(sxy)(sxz) for all x, y, z ∈ M.A morphism of symmetric spaces M, N is a smooth map φ : M → N such that φ(sxy) = sφ(x)φ(y) for all x, y ∈ M. Some basic examples to keep in mind are the following: n (Ex1) the n-dimensional Euclidean space R with symmetries sxy = 2x − y, (Ex2) the n-sphere Sn where the symmetry at x is the restriction to Sn of the axial symmetry about 0x, (Ex3) the hyperbolic plane Π2 with the Riemannian geodesic symmetries, −1 (Ex4) any Lie group G with symmetries sgl = gl g, as well as two solvable two-dimensional examples: 2 0 0 0 0 0 (Ex5) the space A2 = R with symmetries s(a,b)(a , b ) = (2a−a , 2 cosh(a−a )b−b ), 2 0 0 0 0 0 (Ex6) the space B2 = R with symmetries s(a,b)(a , b ) = (2a−a , 2 cos(a−a )b−b ). The connected symmetric spaces in this sense (due to Loos [9]) coincide with the homogeneous symmetric spaces: the homogeneous spaces G/H of connected Lie groups G, with H an open subgroup of the fixed points set Gσ of an involutive automorphism σ of G (called an involution in what follows). The symmetries in −1 that case are given by sgH lH = gσ(g l)H. Such a triple (G, H, σ) will be called a symmetric triple. When H is the full fixed point subgroup Gσ, we simply refer to the symmetric pair (G, σ). A symmetric triple (G, H, σ) is said to realize a symmetric space M if G/H =∼ M as symmetric spaces. Any connected symmetric space M admits a “minimal” realization as a homo- geneous symmetric space of its transvection group. The latter is defined as the subgroup of the automorphism group of M generated by all the products of an even number of symmetries. It is a finite dimensional Lie group which is transitive on M, stabilized by the conjugation (in the automorphism group) by so, for any o ∈ M, and minimal for these properties. A symmetric space is said to be semisimple (respectively, solvable) if its transvection group is semisimple (respectively, solvable). STRONGLY EXPONENTIAL SYMMETRIC SPACES 3 Every Lie group G is a symmetric space when endowed with the symmetries −1 sgl = gl g. It is usually realized by (G × G, Diag(G), σG) with σG(g, l) = (l, g), where G × G is in general larger than the transvection group of G. Given an involution σ on G, two canonical symmetric subspaces of G are defined by P = −1 −1 {g ∈ G | σ(g) = g } and Gσ = {gσ(g) | g ∈ G}. When G is connected, Gσ is the connected component of the identity of P , and is isomorphic to G/Gσ through σ σ −1 the quadratic representation Q : G/G → Gσ : gG → gσ(g) . A symmetric subspace N of a symmetric space M =∼ G/H is said to be normal if there exists a σ-invariant normal subgroup R ⊂ G such that N =∼ R/(R ∩ H) through the isomorphism M =∼ G/H. Then M is said to be a semi-direct product M1 n M2 of two symmetric subspaces M1 and M2 if M is, as a smooth manifold, a direct product M1 × M2, and if M2 is normal in M. On the tangent space m at a given point of a symmetric space M, there is a structure of Lie triple system [·, ·, ·]: m3 → m, generalizing that of Lie algebra for Lie groups. This trilinear bracket satisfies the properties: (Lts1) [X, X, Y ] = 0, (Lts2) [X, Y, Z] + [Y, Z, X] + [Z, X, Y ] = 0, (Lts3) [X, Y, [U, V, W ]] = [[X, Y, U], V, W ] + [U, [X, Y, V ],W ] + [U, V, [X, Y, W ]]. Any Lie algebra g is a Lie triple system with bracket (1) [X, Y, Z] = [[X, Y ],Z], and if G is a Lie group with Lie algebra g, this structure makes it the Lie triple system at e of G seen as a symmetric space. More generally, if σ is an involution on g and if g = m ⊕ h is the corresponding decomposition into (−1) and (+1)-eigenspaces of σ, then the bracket (1) on g restricts to m as a Lie triple system, and any Lie triple system is of that kind. The Lie triple system at the base point eH of a symmetric space G/H coming from a triple (G, H, σ) is given by the bracket (1) on the (−1)-eigenspace m of the involution dσe on Lie(G). Just as any connected symmetric space can be realized as a homogeneous sym- metric space, any Lie triple system can be realized as the (−1)-eigenspace of a Lie algebra involution. The minimal such realization is called the standard embedding and is defined (as a vector space) by g = m ⊕ [m, m] where [m, m] denotes the span of the operators LX,Y = [X, Y, ·] for all X, Y ∈ m. Conditions (Lts1)–(Lts3) show that the following equations define a Lie bracket on g: [X, Y ] = LX,Y , [LX,Y ,Z] = LX,Y Z, [LX,Y ,LU,V ] = LLX,Y U,V + LU,LX,Y V , 4 YANNICK VOGLAIRE for all X, Y, Z, U, V ∈ m. The involution σ on g is − Idm ⊕ Id[m,m] and one sees that (g, σ) is the infinitesimal data at the identity of the tranvection group. On any symmetric space, there is a unique affine connection for which all the symmetries are affine transformations. This connection is complete, torsion-free, and its curvature is parallel. The curvature R at a point x on a symmetric space is related to its Lie triple system at that point by R(X, Y )Z = −[X, Y, Z]. The exponential map at x is denoted Expx : TxM → M. On a symmetric space G/H with Lie triple system m ⊂ g at eH, the set exp m ⊂ G is an open submanifold of P , so that exp m ⊂ Gσ ⊂ P are open inclusions. On a symmetric space with a fixed base point o, there is a product ⊥ defined ∼ by x ⊥ y = sxsoy. If M = G/H, one proves that s X so = exp X ∈ G for all Expo 2 X X ∈ m, so that the product satisfies Expo 2 ⊥ y = exp(X)y. A point z is said to be a midpoint of x and y if szx = y.
Load more

Recommended publications
  • The Pure State Space of Quantum Mechanics As Hermitian Symmetric
    The Pure State Space of Quantum Mechanics as Hermitian Symmetric Space. R. Cirelli, M. Gatti, A. Mani`a Dipartimento di Fisica, Universit`adegli Studi di Milano, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Milano, Italy Abstract The pure state space of Quantum Mechanics is investigated as Hermitian Symmetric K¨ahler manifold. The classical principles of Quantum Mechan- ics (Quantum Superposition Principle, Heisenberg Uncertainty Principle, Quantum Probability Principle) and Spectral Theory of observables are dis- cussed in this non linear geometrical context. Subj. Class.: Quantum mechanics 1991 MSC : 58BXX; 58B20; 58FXX; 58HXX; 53C22 Keywords: Superposition principle, quantum states; Riemannian mani- folds; Poisson algebras; Geodesics. 1. Introduction Several models of delinearization of Quantum Mechanics have been pro- arXiv:quant-ph/0202076v1 14 Feb 2002 posed (see f.i. [20] [26], [11], [2] and [5] for a complete list of references). Frequentely these proposals are supported by different motivations, but it appears that a common feature is that, more or less, the delinearization must be paid essentially by the superposition principle. This attitude can be understood if the delinearization program is worked out in the setting of a Hilbert space as a ground mathematical structure. However, as is well known, the groundH mathematical structure of QM is the manifold of (pure) states P( ), the projective space of the Hilbert space . Since, obviously, P( ) is notH a linear object, the popular way of thinking thatH the superposition principleH compels the linearity of the space of states is untenable. The delinearization program, by itself, is not related in our opinion to attempts to construct a non linear extension of QM with operators which 1 2 act non linearly on the Hilbert space .
    [Show full text]
  • Symmetric Spaces with Conformal Symmetry
    Symmetric Spaces with Conformal Symmetry A. Wehner∗ Department of Physics, Utah State University, Logan, Utah 84322 October 30, 2018 Abstract We consider an involutive automorphism of the conformal algebra and the resulting symmetric space. We display a new action of the conformal group which gives rise to this space. The space has an in- trinsic symplectic structure, a group-invariant metric and connection, and serves as the model space for a new conformal gauge theory. ∗Electronic-mail: [email protected] arXiv:hep-th/0107008v1 1 Jul 2001 1 1 Introduction Symmetric spaces are the most widely studied class of homogeneous spaces. They form a subclass of the reductive homogeneous spaces, which can es- sentially be characterized by the fact that they admit a unique torsion-free group-invariant connection. For symmetric spaces the curvature is covari- antly constant with respect to this connection. Many essential results and extensive bibliographies on symmetric spaces can be found, for example, in the standard texts by Kobayashi and Nomizu [1] and Helgason [2]. The physicists’ definition of a (maximally) symmetric space as a metric space which admits a maximum number of Killing vectors is considerably more restrictive. A thorough treatment of symmetric spaces in general rel- ativity, where a pseudo-Riemannian metric and the metric-compatible con- nection are assumed, can be found in Weinberg [3]. The most important symmetric space in gravitational theories is Minkowski space, which is based on the symmetric pair (Poincar´egroup, Lorentz group). Since the Poincar´e group is not semi-simple, it does not admit an intrinsic group-invariant met- ric, which would project in the canonical fashion to the Minkowski metric ηab = diag(−1, 1 ..
    [Show full text]
  • Holonomy and the Lie Algebra of Infinitesimal Motions of a Riemannian Manifold
    HOLONOMY AND THE LIE ALGEBRA OF INFINITESIMAL MOTIONS OF A RIEMANNIAN MANIFOLD BY BERTRAM KOSTANT Introduction. .1. Let M be a differentiable manifold of class Cx. All tensor fields discussed below are assumed to be of class C°°. Let X be a vector field on M. If X vanishes at a point oCM then X induces, in a natural way, an endomorphism ax of the tangent space F0 at o. In fact if y£ V„ and F is any vector field whose value at o is y, then define axy = [X, Y]„. It is not hard to see that [X, Y]0 does not depend on F so long as the value of F at o is y. Now assume M has an affine connection or as we shall do in this paper, assume M is Riemannian and that it possesses the corresponding affine connection. One may now associate to X an endomorphism ax of F0 for any point oCM which agrees with the above definition and which heuristically indicates how X "winds around o" by defining for vC V0, axv = — VtX, where V. is the symbol of covariant differentiation with respect to v. X is called an infinitesmal motion if the Lie derivative of the metric tensor with respect to X is equal to zero. This is equivalent to the statement that ax is skew-symmetric (as an endomorphism of F0 where the latter is provided with the inner product generated by the metric tensor) for all oCM. §1 contains definitions and formulae which will be used in the remaining sections.
    [Show full text]
  • Symmetric Spaces of Hermitian Type
    Differential Geometry and its Applications 1 (1991) 195-233 195 North-Holland Symmetric spaces of Hermitian type G. ‘Olafsson Mathematisches Znstitut der Uniuersitit Giittingen, Bunsenstraj?e 3-5, D-3400 Giittingen Received 7 February 1991 ‘Olafsson, G., Symmetric spaces of Hermitian type, Diff. Geom. Appl. 1 (1991) 195-233. Abstract: Let M = G/H be a semisimple symmetric space, r the corresponding involution and D = G/K the Riemannian symmetric space. Then we show that the following are equivalent: M is of Hermitian type; r induces a conjugation on D; there exists an open regular H-invariant cone R in q = h’ such that k n 0 # 0. We relate the spaces of Hermitian type to the regular and parahermitian symmetric spaces, analyze the fine structure of D under r and construct an equivariant Cayley transform. We collect also some results on the classification of invariant cones in q. Finally we point out some applications in representations theory. Keywords: Symmetric spaces, semisimple Lie groups, invariant convex cones, causal orientation, ordering, convexity theorem. MS classification: 53635, 57S25, 22E15, 06AlO. Introduction Bounded symmetric domains and their unbounded counterparts, the Siegel domains, have long been an important part of different fields of mathematics, e.g., number theory, algebraic geometry, harmonic analysis and representations theory. So the holomorphic discrete series and other interesting representations of a group live in spaces of holo- morphic functions on such domains. In the last years some interplays with harmonic analysis on affine symmetric spaces have also become apparent, e.g., a construction of non-zero harmonic forms related to the discrete series of such spaces (see [44] and the literature there).
    [Show full text]
  • Symmetric Spaces
    CHAPTER 2 Symmetric Spaces Introduction We remark that a prerequisite for this course is a basic understanding of Rie- mannian geometry, see for instance [Boo86]or[Hel79], which at the same time is a standard reference for the theory of symmetric spaces. See also [Bor98]fora much more condensed version of the latter. In 1926, É. Cartan began to study those Riemannian manifolds,forwhich central symmetries are distance preserving. The term central symmetry is going to be made precise later but you can think of it as a reflection about a point, as in Rn for instance. These spaces are now known as symmetric spaces;henoticedthatthe classification of these is essentially equivalent to the classification of real semisimple Lie algebras. The local definition of symmetric spaces is a generalisation of constant sectional curvature. Just as a differentiable function from R to R is constant if its derivative vanishes identically, one may ask for the Riemanncurvaturetensorto have covariant derivative zero. As incomprehensible as thisstatementmaybeatthe moment, it yields an important equivalent characterizationofsymmetricspaces. Example. The following list of examples of symmetric spaces contains the constant curvature cases. n n (i) Euclidean n-space E =(R ,geucl) has constant sectional curvature zero and isometry group Iso(En) = O(n) !Rn. n n ∼ (ii) The n-sphere S =(S ,geucl) has constant sectional curvature one. Its Riemannian metric arises through restriction of the Riemannian metric of the ambient space Rn+1 Sn to the tangent bundle of Sn.Theisometry group of Sn is O(n). ⊃ (iii) Hyperbolic n-space Hn has constant sectional curvature minus one.
    [Show full text]
  • Symmetric Submanifolds of Symmetric Spaces 11
    SYMMETRIC SUBMANIFOLDS OF SYMMETRIC SPACES JURGENÄ BERNDT Abstract. This paper contains a survey about the classi¯cation of symmetric subman- ifolds of symmetric spaces. It is an extended version of a talk given by the author at the 7th International Workshop on Di®erential Geometry at Kyungpook National University in Taegu, Korea, in November 2002. 1. Introduction A connected Riemannian manifold M is a symmetric space if for each point p 2 M there exists an involutive isometry sp of M such that p is an isolated ¯xed point of M. Symmetric spaces play an important role in Riemannian geometry because of their relations to algebra, analysis, topology and number theory. The symmetric spaces have been classi¯ed by E. Cartan. A thorough introduction to symmetric spaces can be found in [4]. Symmetric submanifolds play an analogous role in submanifold geometry. The precise de¯nition of a symmetric submanifold is as follows. A submanifold S of a Riemannian manifold M is a symmetric submanifold of M if for each point p 2 S there exists an involutive isometry σp of M such that ( ¡X , if X 2 TpS; σp(p) = p ; σp(S) = S and (σp)¤(X) = X , if X 2 ºpS: Here, TpS denotes the tangent space of S at p and ºpS the normal space of S at p. The isometry σp is called the extrinsic symmetry of S at p. It follows from the very de¯nition that a symmetric submanifold is a symmetric space. The aim of this paper is to present a survey about the classi¯cation problem of symmetric submanifolds of symmetric spaces.
    [Show full text]
  • The Symmetric Space for Sln(R)
    The Symmetric Space for SLn(R) Rich Schwartz November 27, 2013 The purpose of these notes is to discuss the symmetric space X on which SLn(R) acts. Here, as usual, SLn(R) denotes the group of n × n matrices of determinant 1. 1 The Group Action The space X denotes the sets of n × n real matrices M such that • M is symmetric: M t = M. • det(M)=1. • M is positive definite: M(v) · v > 0 for all v =6 0. The action of SLn(R) on X is given by g(M)= gMgt. (1) One can easily verify that g(M) ∈ X. Only the third step has anything to it: gMgt(v) · v = M(gt(v)) · gt(v)= M(w) · w> 0. The action we have described really is a group action, because g(h(M)) = g(hMht)gt =(gh)M(gh)t = gh(M). 1 2 The Tangent Space The identity matrix I is the natural origin for X. The tangent space TI (X) consists of derivatives of the form dM V = (0), (2) ds where Ms is a path of matrices in X and M0 = I. Clearly V is a symmetric matrix. Also, to first order, we have Ms = I + sV , and det(I + sV )=1+ s trace(V )+ ... (3) Hence trace(V ) = 0. Conversely, if trace(V ) = 0 and V is symmetric, then I + sV belongs to X to first order. More precisely s2 s3 M = exp(sV )= I + sV + V 2 + V 3 + ··· (4) s 2! 3! gives a path in X, for small s, having V as derivative.
    [Show full text]
  • Symmetric Spaces
    Symmetric Spaces Xinghua Gao May 5, 2014 Notations M Riemannian manifold. N0 normal neighborhood of the origin in TpM. Np normal neighbourhood of p, Np = exp N0. sp geodesic symmetry with respect to p. Φ f dΦf = f ◦ Φ. Φ X dΦX. K(S) sectional curvature of M at p along the section S. r Ds set of tensor fields of type (r; s). I(M) the set of all isometries on M 1 Affine Locally Symmetric Space, Isometry Group and Others Definition 1 (normal neighborhood) A neighborhood Np of p in M is called a normal neighbourhood if Np = exp N0, where N0 is a normal neighborhood of the origin in TpM, i.e. satisfying: (1)exp is a diffeomorphism of N0 onto an open neighborhood Np; (2)if X 2 N0, 0 ≤ t ≤ 1, then tX 2 N0(star shaped). Definition 2 (geodesic symmetry) 8q 2 Np, consider the geodesic t ! γ(t) ⊂ Np passing through p and q s.t. γ(0) = p, γ(1) = q. Then the mapping 0 q ! q = γ(−1) of Np onto itself is called geodesic symmetry w.r.t p, denoted by sp. Remark: sp is a diffeomorphism of Np onto itself and (dsp)p = −I. Definition 3 (Affine locally symmetric) M is called affine locally symmet- ric if each point m 2 M has an open neighborhood Nm on which the geodesic −1 sm sm symmetry sm is an affine transformation. i.e. rX (Y ) = (rXsm (Y )) , 8X; Y 2 X(M). Definition 4 (Pseudo-Riemannian structure) Let M be a C1-manifold.
    [Show full text]
  • Complex Forms of Quaternionic Symmetric Spaces, but It Is Very Close
    Complex forms of Quaternionic Symmetric Spaces Joseph A. Wolf ∗ 19 September 2003 Abstract We give a complete classification of the complex forms of quaternionic symmetric spaces. 1 Introduction Some years ago, H. A. Jaffee found the real forms of hermitian symmetric spaces ([J1], [J2]; or see [HO]).´ That classification turns out to be related to the classification of causal symmetric spaces. This was first observed by I. Satake ([S, Remark 2 on page 30] and [S, Remark on page 87]). Somewhat later it was independently observed by J. Hilgert, G. Olafsson´ and B. Ørsted; see [HO],´ especially Chapter 3 and the Notes at the end of that Chapter. I learned about that from Bent Ørsted. He and Gestur Olafsson´ had informally discussed complex forms of quaternionic symmetric spaces and found examples for the classical groups, for G2 , and perhaps for F4 . Ørsted told me about the classical ones, and we rediscovered examples for G2 and F4 . I thank Bent Ørsted for agreeing to my incorporating those examples into this note. Later I used the computer program LiE [L] to find examples for E6 , E7 and E8 . In this note I write down a complete classification for complex forms L/V of quaternionic symmetric spaces G/K. The definitions and some preliminary results are in Sections 2 and 3, the main results are stated in Section 4, and the proofs are in Sections 5, 6, 7 and 8. The case where G is a classical group and rank(L) = rank(G) is handled, essentially by matrix considerations, in Section 5. That, of course, does not work comfortably for the exceptional groups, which must be approached by means of their root structure.
    [Show full text]
  • [PDF] Chapter 12: Riemannian Symmetric Spaces
    12 Riemannian Symmetric Spaces Contents 12.1 Brief Review 213 12.2 Globally Symmetric Spaces 215 12.3 Rank 216 12.4 Riemannian Symmetric Spaces 217 12.5 Metric and Measure 218 12.6 Applications and Examples 219 12.7 Pseudo Riemannian Symmetic Spaces 222 12.8 Conclusion 223 12.9 Problems 224 In the classification of the real forms of the simple Lie algebras we en- countered subspaces p, ip on which the Cartan-Killing inner product was negative-definite (on p) or positive-definite (on ip). In either case these subspaces exponentiate onto algebraic manifolds on which the invariant metric gij is definite, either negative or positive. Manifolds with a defi- nite metric are Riemannian spaces. These spaces are also globally sym- metric in the sense that every point looks like every other point — because each point in the space EXP (p) or EXP (ip) is the image of the origin under some group operation. We briefly discuss the properties of these Riemannian globally symmetric spaces in this chapter. 12.1 Brief Review In the discussion of the group SL(2; R) we encountered three symmet- ric spaces. These were S2 SU(2)/U(1), which is compact, and its 2 ∼ dual H2+ = SL(2; R)/ SO(2) = SU(1, 1)/U(1), which is the upper sheet of the two-sheeted hyperboloid. ‘Between’ these two spaces oc- 2 curs H1 = SL(2; R)/SO(1, 1), which is the single-sheeted hyperboloid. These spaces are shown in Fig. 12.1. The Cartan-Killing inner product in the linear vector subspace su(2) − u(1) is negative definite.
    [Show full text]
  • Representations of Locally Symmetric Spaces
    IC/95/284 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS REPRESENTATIONS OF LOCALLY SYMMETRIC SPACES M. S. Rahman INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION MIRAMARE-TRIESTE 1 V0L 27^07 IC/95/284 1. Introduction International Atomic Energy Agency There is a class of Riemannian manifolds, the symmetric spaces, which plays a very and important role in Differential Geometry. United Nations Educational Scientific and Cultural Organization A symmetric space is a Riemannian manifold whose curvature tensor is invariant under INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS all parallel translations. The theory of symmetric spaces was initiated by E. Cartan in 192G, and was vigorously REPRESENTATIONS OF LOCALLY SYMMETRIC SPACES developed and classified completely by him by a reduction to the classification of simple Lie algebras in the late 1920s. In spirit to their definition, symmetric spaces form a special topic in Riemannian geometry. M. S. Rahman1 This theory leads to the remarkable fact that symmetric spaces are locally just the International Centre for Theoretical Physics, Trieste, Italy. Rienmnnian manifolds of the form ABSTRACT where ft" is a Euclidean ri-space, G is a semi-simple Lie group which has an involutive automorphism with fixed point set as the (essentially) compact group A', ami -j? is treated Locally symmetric spaces in reference to globally and Hermitian symmetric Rieman- with a G-invariant Riemannian structure. nian spaces arc studied. Some relations between locally and globally symmetric spaces The theory of symmetric spaces helps to unify and explain iu a general way various are exhibited. A lucid account of results on relevant spaces, motivated by fundamental phenomena in classical geometries.
    [Show full text]
  • Isometries Between Quantum Symmetric Spaces
    ISOMETRIES BETWEEN NON-COMMUTATIVE SYMMETRIC SPACES ASSOCIATED WITH SEMI-FINITE VON NEUMANN ALGEBRAS PIERRE DE JAGER AND JURIE CONRADIE Abstract. In this article we show that positive surjective isometries between symmetric spaces associated with semi-finite von Neumann algebras are pro- jection disjointness preserving if they are finiteness preserving. This is sub- sequently used to obtain a structural description of such isometries. Further- more, it is shown that if the initial symmetric space is a strongly symmetric space with absolutely continuous norm, then a similar structural description can be obtained without requiring positivity of the isometry. 1. Introduction The form of isometries between Lp-spaces was first described by Banach (in the case of finite measure spaces ([1])) and Lamperti (for σ-finite measure spaces ([18])). In the proofs of these results essential use is made of the fact that isometries map functions with disjoint support to functions with disjoint support. Representations of isometries between more general symmetric function spaces were obtained by Zaidenberg ([23]). We will define symmetric spaces below, but mention that well- known examples of such spaces include the Lp, Orlicz and Lorentz function spaces. A detailed account of results on isommetries in the commutative settings and the techniques used in the proofs can be found in [13]. Non-commutative symmetric spaces are Banach spaces of closed, densely-defined operators affiliated with a von Neumann algebra. In the special case where the un- derlying von Neumann algebra is commutative, and hence isometrically isomorphic to an L∞ space over some localizable measure space, we obtain the commutative (classical) symmetric function spaces.
    [Show full text]